Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Advertisements
उत्तर
Let I = `int sqrt(x^2 + 2x + 5).dx`
= `int sqrt(x^2 + 2x + 1 + 4)dx`
= `int sqrt((x + 1)^2 + 2^2).dx`
= `((x + 1)/2) int sqrt((x + 1)^2 + 2^2) + 2^2/(2)log|(x + 1) + sqrt((x + 1)^2 + 2^2)| + c`
= `((x + 1)/2)sqrt(x^2 + 2x + 5) + 2log|(x + 1) + sqrt(x^2 + 2x + 5)| + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x2 log x.
Integrate the function in (sin-1x)2.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/("9x"^2 - 25)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int sin4x cos3x "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int_0^1 x tan^-1 x dx` = ______.
`int1/sqrt(x^2 - a^2) dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
